Gravity field of the earth - Physical Geodesy - page 62

exact ellipsoid, the gravity field of an ellipsoid is of fundamental practical
importance because it is easy to handle mathematically and the deviations
of the actual gravity field from the ellipsoidal “normal” field are so small that
they can be considered linear. This splitting of the earth's gravity field into a
“normal” and a remaining small “disturbing” field considerably simplifies the
problem of its determination; the problem could hardly be solved otherwise.
Therefore, we assume that the normal figure of the earth is a level ellip-
soid, that is, an ellipsoid of revolution which is an equipotential surface of
a normal gravity field. This assumption is necessary because the ellipsoid is
to be the normal form of the geoid, which is an equipotential surface of the
actual gravity field. Denoting the potential of the normal gravity field by

U = U ( x, y, z ) , (2-96)

we see that the level ellipsoid, being a surface U = constant, exactly corre-
sponds to the geoid, which is defined as a surface W = constant.
The basic point here is that by postulating that the given ellipsoid be
an equipotential surface of the normal gravity field, and by prescribing the
total mass M , we completely and uniquely determine the normal potential
U . The detailed density distribution inside the ellipsoid, which produces the
potential U , is quite uninteresting and need not be known at all. In fact,
we do not know of any “reasonable” mass distribution for the level ellipsoid
(Moritz 1990: Chap. 5). Pizzetti (1894) unsuccessfully used a homogeneous
density distribution combined with a surface layer of negative density, which
is quite “unnatural”.
This determination is possible by Dirichlet's principle (Sect. 1.12): The
gravitational potential outside a surface S is completely determined by know-
ing the geometric shape of S and the value of the potential on S . Originally
it was shown only for the gravitational potential V , but it can be applied to
the gravity potential

U = V + 2 ω 2 ( x 2 + y 2 ) (2-97)

as well if the angular velocity ω is given. The proof follows that in Sect. 1.12,
with obvious modifications. Hence, the normal potential function U ( x, y, z )
is completely determined by

1. the shape of the ellipsoid of revolution, that is, its semiaxes a and b ,
2. the total mass M ,and
3. the angular velocity ω .

The calculation will now be carried out in detail. The given ellipsoid S 0 ,

x 2 + y 2
a 2

+ z 2
b 2

=1 , (2-98)